Abstract
We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M= G/ K of compact type with rk G−rk K⩽1. Let ḡ be another metric with scalar curvature κ ̄ , such that g ̄ ⩾g on 2-vectors. We show that κ ̄ ⩾κ everywhere on M implies κ ̄ =κ . Under an additional condition on the Ricci curvature of g, κ ̄ ⩾κ even implies g ̄ =g . We also study area-non-increasing spin maps onto such Riemannian manifolds.
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